Solar-like Oscillations in other Stars or The only way to test directly stellar structure theory I.Scaling Relations II. Results

Kjeldsen & Bedding, 1995, A&A, vol. 293, 87 „Asteroseismology of solar-like stars has so far produced disappointing results. Despite repeated attempts and ever- increasing sensitivity, there have been no unambiguous detection of solar-like oscillations on any star except the Sun.“ Asteroseismology: A relatively new field Solar-like oscillations: Any oscillation that is excited by convection in the outer part of the star analogous to the 5 minute oscillations.

Only low-degree modes can be detected Unlike the sun for which we have detected mode with l up to 400, this is because the Sun is resolved. For stars for which we measure integrated light, there is a high degree of cancellation. We can only detect low degree modes, l = 0-3

n, l =   (n + l /2 +  – l ( l +1 )D 0  0 = large spacing D 0 is sensitive to the sound speed near the core  is sensitive to the surface layers The last term is related to the small frequency spacing The Asymptotic Limit for p-mode Oscillations For high order modes: n >> l Note: l = 0 → radial pulsations. For pure radial modes the frequency spacing is the large spacing

If there is no small frequency spacing n+1, 0 = n,2 but the gradient of the sound speed (small frequency spacing) lifts this degeneracy Definition:  02 is the small separation as the frequency spacing of adjacent modes with l = 0 and l =2  13 is the small separation as the frequency spacing of adjacent modes with l = 1 and l =3  01 is the amount modes l = 1 are offset between the mid- point of l =0 modes on either side. If the asymptotic relation holds exactly then: D 0 =  02 6  01 2 == 10

The Solar Power Spectrum showing the large and small spacings:  01

l=0l=1l=2  A Diagnostic Tool: Echelle Diagrams n, l = 0 + k  = a reference k = integer 1 = 0 →  Echelle diagrams take advantage of the fact that in the asymptotic relationship p-modes are equally-spaced in frequency. An echelle diagram basically cuts the frequency axis into chunks of  and stacks them on each other

0 11  =  0 /2) 0 0 l =0 l =1 l =2 Radial order n, increasing l =3  02  13 Modulo Frequency Frequency

The „Asteroseismic“ H-R Diagram The goal of asteroseismology is to detect enough modes to derive the large and small frequency spacings. From these you get the mass and age of the star from the „Asteroseismic“ H-R diagram

Estimating the Photometric Amplitude of Oscillations (Kjeldsen & Bedding, 1995, 293, 87) LL L ∝ ( ( bol TT T The change in luminosity of the star is due almost entirely to temperature changes TT T ∝   For adiabatic oscillations   = v cscs To first order the density compression for an adiabatic sound wave

Estimating the Photometric Amplitude of Oscillations The adiabatic sound speed: c s 2 =  ln P  ln  [ [ ad PP   ln P  ln  = [ [ ad 5/3 Ideal gas law give:  P ∝  T cs2∝ Tcs2∝ T

Estimating the Photometric Amplitude of Oscillations ~ LL L ( ( bol v osc √T eff Assume T ≈ T eff This expresses the luminosity amplitude of stellar oscillations in terms of the velocity amplitude. Note: this is for small amplitude variations.

In comparing to observations, which are made at specific wavelengths we have to take into account that the luminosity amplitude of an oscillation depends on the wavelength that it is observed. LL L ( ( bol = LL L ( ( bol ( ( ~ LL L ( ( bol ( ( = 623 nm T eff /5777K Combining: LL L ( ( = v osc T eff –1.5

When comparing to real data a better fit is obtained using an exponent of –2 instead of –1.5. This is not unsurprising since we used an adiabatic approximation so there may be temperature corrections when comparing to real data. The revised equation becomes: LL L ( ( = v osc / m s –1 (T eff / 5777 K) 2 ( / 550 nm) 20.1 ppm LL L ( ( bol = v osc / m s –1 (T eff / 5777 K) ppm 1 ppm = 1 part per million (10 –6 ) =  mag

This shows the predicted luminosity variations versus observed variations for a variety of pulsating stars. The previous expression is a good approximation even when the oscillations are nonlinear or non-adiabatic

Comparison of the predictions for this simple expression to model predictions from Christensen-Daalsgard & Frandsen (1983, Sol. Physics, 82, 469)

Scaling the Velocity Amplitude to other stars V osc ~ L/M Stellar models suggest that the velocity amplitude scales: We can thus take the solar amplitudes and scale these according the values to other stars Estimating the Velocity Amplitude

Amplitude Scaling Laws v osc = L/L סּ M/M סּ (23.4 ± 1.4) cm/sec LL L ( ( = L/L סּ (T eff / 5777 K) 2 ( / 550 nm) (4.7 ± 0.3) ppm (M/M סּ ) These equations scale to values observed for the sun

max ≈ 3000  Hz  0 ≈ 135  Hz How do we scale these to other stars? The Solar Power Spectrum

l = 0 l = 1 n max n max +1 00 11 Each peak in the frequency spectrum corresponds to a harmonic mode characterized by a radial order n, and an angular degree l. For stars observed in integrated light we most likely detect only l = 0,1

nl ≈   (n + l/2 +  )  ~ 1.6 for the sun  0 ≈ [2 dr/c s ] –1 In other words the large spacing is the inverse travel time of a sound wave passing directly through the star. It is  Hz for the Sun (i.e. about 2 hours travel time) ∫ R 0 c s is the sound speed

This sound travel time is related to the global dynamical Timescale of the star: d2Rd2R dt2dt2 = GM R2R2 – = G  R M sun = 2 ×10 33 gm R sun = 7 ×10 10 cm  = mean density t = (G  ) –½  = 1.4 gm/cm 3 The dynamical time for the sun is about 1 hour. This would be the period of radial pulsations, if they were present. Or, if you were to hit the sun, this is the fastest it could respond

 0 ≈ [2 dr/c s ] –1 ∫ R 0 Where T is now the average internal Temperature ≈ cscs R ∝ √ T R cs2∝ Tcs2∝ T But recall that the adiabatic sound speed satisfies:

Footnote: Equation for Hydrostatic Equilibrium 1. Hydrostatic Equilibrium P + dP P dA r + dr M(r) The gravity in a thin shell should be balanced by the outward gas pressure in the cell r dm A P +dP dr P Gravity

F p = PdA –(P + dP)dA = –dP dA Pressure Force F G = – GM(r) dM r2r2 Gravitational Force r 0 M(r) = ∫  (r) 4  r 2 dr dM =  dA dr Both forces must balance: F P + F G = 0 dPdP drdr = – G  (r)M(r) r2r2

Some more approximations: dP dr = – GM r2r2  Stellar structure equation for hydrostatic equilibrium dP dr = – GM R2R2  R3R3 P R = – GM 2 R5R5 P ∝P ∝ M2M2 R4R4 Ideal gas law:  P ∝  T T M2M2 R4R4 ∝  R3R3 T ∝  R

00 √ T R ∝ T ∝  R ½ 00 M R3R3 ∝ ( ) ∝  ½ The frequency splitting is directly proportional to the square root of the mean density of the star! 0 ≈0 ≈ (M/M סּ ) 1/2 (R/R סּ ) 3/2  Hz

The power has an envelope whose maximum is at max ≈ 3mHz For the sun. The shape of the envelope and value of max is determined by excitation and damping. The acoustic cutoff, ac, defines a dynamical timescale for the atmosphere, so we expect max to scale as ac ac ∝ c s /H p For frequencies above the acoustic cutoff the energy of the mode decreases exponentially with height in the atmosphere And what is the scale height of the atmosphere?

dP = –g  dh  F = g  A h A F/A = g  h Pressure gravity  = P  /kT dP = gg kT – P dh P = P o exp (  gh kT ) – = P o exp ( h H ) –  =  o exp ( h H ) – Scale height H = kT/  g Footnote #2 :Scale Height of Atmosphere h F = GMm/R 2

H p = kT/  g H p is the pressure scale height where the pressure decreases exponentially: P = P o e –h/H p cs ∝cs ∝ √T√T max ∝ g /√ T ∝ M R2R2 √T√T max = M/M סּ (R/R סּ ) 2 √T eff /5777K 3.05 mHz max  ∝ ac ∝ c s /H p

The maximum power in the sun is seen for modes with n ≈ 21 n max = M/M סּ (R/R סּ ) 2 √T eff /5777K X 22.6 – 1.6 () ½

Summary of Scaling Relationships v osc = L/L סּ M/M סּ (23.4 ± 1.4) cm/sec LL L ( ( = L/L סּ (T eff / 5777 K) 2 ( / 550 nm) (4.7 ± 0.3) ppm (M/M סּ ) max = M/M סּ (R/R סּ ) 2 √T eff /5777K 3.05 mHz 0 ≈0 ≈ (M/M סּ ) 1/2 (R/R סּ ) 3/2  Hz n max = M/M סּ (R/R סּ ) 2 √T eff /5777K X 22.6 – 1.6 ( ) ½

Scaling between  and max Stello et al. 2009, MNRAS, 400, L80  0 = (0.263 ± 0.009)  Hz ( max /  Hz) 0.772±0.005

The previous expression is useful when you do not have enough data to derive the large spacing. Fit a Gaussian to the envelope of excess power, find max and compute  0 max ≈ 950  Hz →  0 ≈ 52  Hz. This is data for Procyon, and as we shall see this is near the correct value

Stellar Oscillations (or not) of Procyon Procyon is a bright (m v = 0.36) star in the winter sky that is slightly evolved off the main sequence Spectral Type: F5 IV Teff = 6450 K Mass = 1.42 M סּ Radius = 2.07 R סּ L = 7.03 L סּ V osc = 0.8 m/s  L/L ≈ 2 x 10 –5  0 ≈ 54  Hz max ≈ 1 mHz P ≈ 17 min n max ≈ 11

First possible detections of p-mode oscillations with radial velocity measurements were made with a fiber fed spectropgraph: FOE

Martic et al found convincing evidence for p-mode oscillations in Procyon using ELODIE and the simultaneous Th-Ar for radial velocity measurements Excess power is in the same frequency range as found by Brown et al. Most probable large frequency spacing ≈ 55  Hz More convincing evidence…

2-site campaign using ELODIE and AFOE Even more convincing evidence… but…

Microvariability and Oscillations of STars 1 MOST is a 15cm telescope (Canada‘s First Space Telescope) designed to study stellar oscillations. It can continuosly observe stars for up to 60 days. In 2004 MOST observed Procyon for 32 days. 1 PI: Jaymie Matthews, also known as Matthew‘s Own Space Telescope

Expected power of oscillations at 1 mHz MOST found no evidence of solar-like oscillations in the photometry of Procyon casting doubt on the radial velocity results.

Top panels: simulated power spectra of oscillations for Procyon and with 3 time scales for the mode lifetimes. In the lower panel nose has been added to the simulated data to reach the noise level of MOST. Conclusion #1: MOST could not have detected the pulsations even if they were present

Bedding et al. A&A, 432, L43, 2005 The „power density“ of the MOST observations is significantly higher than for the EW measurements of Kjeldsen et al. 1999, and for the Sun. Conclusion #2, the noise level of MOST is too high.

A multi-site campaign from 9 observatories was conducted on Procyon in 2008 Above: the radial velocity measurements from the various sites (black: TLS). Left: preliminary power spectrum

Stellar Oscillations  Boo Spectral Type: G0 IV m v ´= 2.68 T eff = 6050 K Mass = 1.6 M סּ Radius = 2.8 R סּ L = 9.5 L סּ V osc = 1.4 m/s  L/L ≈ 2.5 x 10 –5 (25 ppm)  0 ≈ 36  Hz max ≈ 0.61 mHz P ≈ 27 min n max ≈ 8

Looking for oscillations through temperature changes L ∝ T 4 (assuming constant radius).  L/L ∝ 4  T/T. The strength of the Balmer lines of hydrogen are sensitive to temperature and the variations are expected to be 6 ppm. Advantage: you do not need a high resolution spectrograph to measure the strength of the hydrogen lines!

Power Spectrum of Temperature Variations of  Boo

Echelle diagram based on equivalent width (temperature) variations of Hydrogen lines Echelle diagram combining temperature and radial velocity measurements

Stellar Oscillations  Hya Spectral Type: G2 IV T eff = 5774 K Mass = 1.1 M סּ Radius = 1.87 R סּ L = 3.53 L סּ V osc ≈ 0.8 m/s  L/L ≈ 15 ppm  0 ≈ 55  Hz max ≈ 1 mHz P ≈ 17 min n max ≈ 11

Radial velocity measurements from a multi-site campaign (HARPS + UCLES). Black are HARPS, red are UCLES

V osc ≈ 0.8 cm/s

Stellar Oscillations  Cen A Spectral Type: G2 IV T eff = 5810 K Mass = 1.1 M סּ Radius = 1.22 R סּ L = 1.5 L סּ V osc = 0.3 m/s  L/L ≈ 6.3 ppm  0 ≈ 105  Hz max ≈ 2.2 mHz P ≈ 7.5 min n max ≈ 11

Radial velocity measurements taken with an iodine cell: Power spectrum

Bouchy & Carrier 2003 l = 1 l = 0 Power spectrum of data taken with a different instrument shows power at the right frequency range  0 ≈ mHz l = 2

Radial Velocity Measurements of a Cen A with HARPS Power Spectrum Bazot et al. 2007, A&A, 470, 295

Large separation as a function of frequency for l =0,1,2 modes Small separation as a for l =0 Bazot et al. 2007, A&A, 470, 295

And a questionable claim of rotational splitting: n l m ≈ m, l ± m  For l =2 This gives  ≈ 1 mHz → P = 11.5 days, but estimated rotational period is 28.8 days

Stellar Oscillations  Cen B Spectral Type: K1 V m v = 1.33 T eff = 5260 K Mass = 0.90 M סּ Radius = 0.86 R סּ L = 0.5 L סּ V osc = 0.13 m/s  L/L ≈ 3 ppm  0 ≈ 160  Hz max ≈ 3.9 mHz P ≈ 4.3 min n max ≈ 20

The Radial Velocity and Power Spectrum of a Cen B

Eggenberger et al. 2004, A&A, 417, 235

Stellar Oscillations  Ara Spectral Type: G3 IV-V T eff = 5813 K Mass = 1.1 M סּ Radius = 1.3 R סּ L = 1.91 L סּ V osc = 0.4 m/s  L/L ≈ 8 ppm  0 ≈ 95  Hz max ≈ 2 mHz P ≈ 8.4 min n max ≈ 17 [Fe/H] = +0.32x metals as the sun

A 14 M earth planet around  Ara was discovered as part of an asteroseismic run.

RV variation with long term variation due to planet Short time segment showing oscillations Full data set

l = 2 l = 0 l = 1 l = 3  0 = 90  Hz Power spectrum

These are stars with metallicity [Fe/H] ~ +0.3 – +0.5 There is believed to be a connection between metallicity and planet formation. Stars with higher metalicity tend to have a higher frequency of planets. Valenti & Fischer The Planet-Metallicity Connection

Two scenarios have been proposed to explain the high metallicity of planet hosting Scenario 1: The high metal content is primordial and reflects the abundance in the star and thus the proto-planetary disk. A higher metal content implies that the planets are easier to form (core accretion theory) → the high metal abundance forms more planets Scenario 2: The high metal content is only on the surface layers of the star and result from the accretion of planetary bodies onto the star → the planets cause the high metalic abundance The two different scenarios should produce different asteroseismic (acoustic) signals.

Over-metallic star Accretion model A slightly better fit is provided with the accretion model, however the main difference is the cross-over of the l =0,2 modes at 2.5 mHz. Unfortunately, this is beyond the highest frequency of the observations. One day asteroseismology may provide the answer to the planet-metallicity effect.

Stellar Oscillations Ind Spectral Type: G0III T eff = 5300 K Mass = 0.9 M סּ Radius = 3 R L = 5.5 L סּ V osc = 1.4 m/s  L/L ≈ 34 ppm  0 ≈ 25  Hz max ≈ 0.3 mHz P ≈ 55 min n max ≈ 6

 o =  Hz Stellar Oscillations Ind

Best fits to the Teff, Radius, Mass, and Age of Ind from model fitting to the observed frequencies

What about Main Sequence stars? Stellar Oscillations  Cet Spectral Type: G8 V T eff = ~5400 K Mass = 0.9 M סּ Radius = 0.79 R סּ (interferometry) L = 0.49 L סּ V osc = 0.13 m/s  L/L ≈ 3 ppm  0 ≈ 178  Hz max ≈ 4.4 mHz P ≈ 3.8 min n max ≈ 31 Note:  Cet is often used as a radial velocity standard star by planet search programs

HARPS data

Always have a control star! Instrumental effects!

Power spectra after correcting instrumental effects

Echelle Diagram for  Cet

Interferometry gives a radius of 0.79 R סּ. From the large spacing one gets a mass of ± M סּ good to 1.6%

Comparison of observed max to those predicted by the scaling relationships:

The „Asteroseismic“ H-R Diagram  Cen A  Cen B  Boo Procyon  Hya The models of course need refinement but one can say that most of these stars are evolved.  Ara  Cet

Asteroseismic Targets Most asteroseismic targets have been evolved stars because these produce the highest amplitudes.

Also showing solar-like (p-mode) oscillations, but will be discussed later: rapidly oscillating Ap stars K giant stars solar-like oscillations from Space Missions: expected amplitudes are 5-20 ppm. Such precision can only be obtained from space. Space also offers the possibility for continuous coverage

Stellar Oscillations Network Group SONG plans a series of 1-m telescopes equipped with spectrographs + iodine absorption cells and spread across the globe.

Summary 1. Scaling relationships work remarkably well for predicting the amplitude and frequencies of solar-like oscillations in other stars over a wide range of amplitudes, periods, spectral types, etc. 2.About a dozen solar-type stars have been studied with stellar oscillations using ground based observations. These have exclusively used the radial velocity method. Photometric amplitudes are expected to be ≈ 10 –5 → you need to go into space 3. From the ground multi-site campaigns are the most effective means of studying stellar oscillations 4. Asteroseismology it the best means of deriving the mass, radius, effective temperature, helium and heavy element fraction and internal structure of the star.